In 1966, R. A. Wijsman studied some optimum properties of the sequential probability ratio test, he considered a mode of convergence for sequences of closed convex sets in R^n. Since then, this type of convergence has attracted the attention of both analysts and topilogists, and its applications in convex analysis and Banach space geometry have been explored. Despite of investigation on Wijsman convergence in the past 40 years, some fundamental questions concerning its topology remain unsolved. For example, when does the Wijsman topology have the Baire property? When is the Wijsman topology normal? To attack these questions, the techniques of splitting and embedding have been employed. In this talk, I shall highlight recent progress towards these questions. In particular, some partial solutions and my recent joint work with H. J. K. Junnila , A. H. Tomita et al will be presented.